Summary: NEST ALGEBRAS IN c1
by
Alvaro Arias*
ABSTRACT. In this paper we address some basic questions of the Banach space
structure of the nest algebras in the trace class in particular, we study whether any two
of them are isomorphic to each other, and show that the nest algebras in the trace class
have bases. We construct three non-isomorphic examples of nest algebras in c1 present a
new proof of the primarity of c1 (Arazy, Ar1], Ar2]), and prove that K(H), and the nest
algebras in B(H) are primary.
1. INTRODUCTION.
In the present paper we study some basic questions of the Banach space structure
of the nest algebras. In particular, we study whether any two nest algebras in c1 are
isomorphic to each other.
The answer to this question is known for the other Schatten p-classes, cp, and for
B(H): All the nest algebras in cp, 1 < p < 1 are isomorphic to cp. This is an easy
consequence of the results of Macaev Ma] and Gohberg and Krein GK] that say that the
nest algebras in cp, 1 < p < 1 are complemented in cp. Likewise, all the nest algebras in
B(H) are completely isomorphic to each other (see A2]).
The structure of the nest algebras in c1 is richer. We will show, for instance, that if
the complete nest is uncountable, then the nest algebra in c1 is isomorphic to the contin-